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Preprint Number 1749

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1749. JinHoo Ahn, Joonhee Kim
SOP1, SOP2, and antichain tree property

Submission date: 24 March 2020


In this paper we study some tree properties and their related indiscernibilities. Firstly we prove that SOP2 can be witnessed by a formula with a tree of tuples holding `arbitrary homogeneous inconsistency' (e.g. weak k-TP1 conditions or other possible inconsistency cofigurations). Secondly we introduce a notion of tree-indiscernibility which preserves witnesses of SOP1 and by using this, we investigate the problem of (in)equality of SOP1 and SOP2. As is well-known SOP2 implies SOP1 but it is not known whether the converse holds. We introduce the notions of antichain tree property (ATP) and 1-strictly strong order property (SSOP1). We show that the two notions are in fact equivalent, and implying SOP1, but the converse does not hold. And we show that they must appear in any theory (if exists) having SOP1 but not SOP2 (NSOP2). Lastly, we construct a structure (a prototype example having ATP) withnessing SOP1-NSOP2 in the formula level, i.e., there is a formula having SOP1, while any finite conjunction of it does not witness SOP2 (but a variation of the formula still has SOP2).

Mathematics Subject Classification: 03C45

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Full text arXiv 2003.10030: pdf, ps.

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